Argand diagram transformations

<p>The above data have new names in the context of an Argand diagram, which we summarize below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Vectorial data</th><th style="text-align: left; --cell-justify: flex-start">New name in Argand diagram</th><th style="text-align: left; --cell-justify: flex-start">Notation</th><th style="text-align: left; --cell-justify: flex-start">Formula</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Position vector of \(Z\)</td><td style="text-align: left; --cell-justify: flex-start">Complex number</td><td style="text-align: left; --cell-justify: flex-start">\(z = x + yi\)</td><td style="text-align: left; --cell-justify: flex-start">N. A.</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Horizontal component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\)</td><td style="text-align: left; --cell-justify: flex-start">Real part</td><td style="text-align: left; --cell-justify: flex-start">\(\mathrm{Re}(z)\)</td><td style="text-align: left; --cell-justify: flex-start">\(x\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Vertical component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\)</td><td style="text-align: left; --cell-justify: flex-start">Imaginary part</td><td style="text-align: left; --cell-justify: flex-start">\(\mathrm{Im}(z)\)</td><td style="text-align: left; --cell-justify: flex-start">\(y\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Length of \(\overrightarrow{OZ}\)</td><td style="text-align: left; --cell-justify: flex-start">Modulus</td><td style="text-align: left; --cell-justify: flex-start">\(|z| = r\)</td><td style="text-align: left; --cell-justify: flex-start">\(\sqrt{x^2 + y^2}\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Angle made with positive \(x\)-axis</td><td style="text-align: left; --cell-justify: flex-start">Argument</td><td style="text-align: left; --cell-justify: flex-start">\(\arg(z) = \theta\)</td><td style="text-align: left; --cell-justify: flex-start">Not so simple</td></tr></tbody></table></div></div>

The above data have new names in the context of an Argand diagram, which we summarize below.

Modulus-argument or polar form

Modulus-argument or polar form

Vectorial data	New name in Argand diagram	Notation	Formula
Position vector of \(Z\)	Complex number	\(z = x + yi\)	N. A.
Horizontal component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\)	Real part	\(\mathrm{Re}(z)\)	\(x\)
Vertical component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\)	Imaginary part	\(\mathrm{Im}(z)\)	\(y\)
Length of \(\overrightarrow{OZ}\)	Modulus	\(\|z\| = r\)	\(\sqrt{x^2 + y^2}\)
Angle made with positive \(x\)-axis	Argument	\(\arg(z) = \theta\)	Not so simple